Teaching neural network on fixed data
The problem you describe is a matter of perspective; although you want to predict the inputs, defining the outputs of the BB (blackbox) as the inputs to a new system (an inverse BB) whose outputs are that of the inputs of the original BB, then one can use previous literature. There are some assumptions using this method such as "each input can be calculated from an inverse function using 1 or more outputs" so take precaution.

I'd recommend reading through this paper by Richard Sutton. Its introduction defines the problem mathematically and references some common approaches as well as its own method.